Raymond Flood - Mathematics - Great Mathematicians, Great Mathematics (2014-16)

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source: GreshamCollege       2014年9月18日
The transcript and downloadable versions of the lectures are available from the Gresham College Website: http://www.gresham.ac.uk/lectures-and...
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1:01:34 Fermat's Theorems 
Gresham Professor of Geometry, Raymond Flood, begins his series 'Great Mathematicians, Great Mathematics' with Pierre de Fermat:http://www.gresham.ac.uk/lectures-and...
The seventeenth century mathematician Pierre de Fermat is mainly remembered for contributions to number theory even though he often stated his results without proof and published very little. He is particularly remembered for his ‘last theorem’ which was only proved in the mid-1990s by Andrew Wiles. He also stated other influential results, in particular Fermat’s ‘Little Theorem’ about certain large numbers which can be divided by primes. His ‘Little Theorem’ is the basis of important recent work in cryptography and internet security.
51:00 Newton's Laws 
An explanation of Newton's mathematics and its applications today: http://www.gresham.ac.uk/lectures-and...
In his Principia Isaac Newton used his law of universal gravitation and three laws of motion to explain elliptical planetary motion, the orbits of comets, the variation of the tides and the flattening of the earth at its poles. Further important work on celestial mechanics was undertaken by Joseph-Louis Lagrange, Pierre-Simon Laplace and Henri Poincaré. It was Poincaré who discovered that even with Newton’s deterministic laws the resulting motion may be irregular and unpredictable, the basis of modern day chaos theory.
50:57 Euler's Exponentials 
A thorough examination of the life and work of one of histories greatest mathematicians, the "Shakespeare of Numbers", Leonhard Euler: http://www.gresham.ac.uk/lectures-and...
Leonhard Euler was the most prolific mathematician of all time. He introduced the symbols e for the exponential number f for a function and i for √-1. He discovered what many mathematicians consider to be the most beautiful expression in mathematics, e ix = cosx + i sinx: a relation connecting the exponential and trigonometric functions. The exponential function and its inverse the logarithm function appear throughout mathematics and its applications, in physics, engineering, mathematical biology, chemistry and economics.
53:08 Fourier's Series
The life an work of one of the greatest mathematicians examined in detail: http://www.gresham.ac.uk/lectures-and...
Joseph Fourier was interested in the mathematical study of the diffusion of heat in solid bodies which he described using infinite trigonometric series which are now known as Fourier series. These series had major applications in many other types of physical problems and led to many of the most important mathematical discoveries of the nineteenth century. Fourier series are used not only in engineering, geology and astronomy but also in number theory, control theory and statistics.
57:09 Möbius and his Band 
A history of one of the greatest mathematicians contribution to our understanding of the world: http://www.gresham.ac.uk/lectures-and...
Many people have heard of the Möbius band, a one-sided surface, but the work of August Möbius was more far reaching than just inventing a topological curiosity. His work in geometry, celestial mechanics and topology, sometimes called rubber sheet geometry, illuminates the mathematical and astronomical life of the nineteenth century. Möbius’s concerns, concepts and the methods he helped to develop played an important part in twentieth century mathematics.
53:29 Cantor's Infinities 
Although many people contributed to the study of infinity over the centuries it was Georg Cantor in the nineteenth century who established its modern development. Cantor created modern set theory and established the importance of one-to-one correspondence between sets. For example he showed that the set of all integers can be put into one-to-one correspondence with the set of all fractions and so these two sets have the same infinity.  But he also proved the remarkable result that there are infinitely many infinities, all of different sizes.
49:59 Einstein's Annus Mirabilis, 1905 
Professor Flood reviews the year that made Einstein famous as he published some of his greatest work: http://www.gresham.ac.uk/lectures-and...
In 1905, his 'year of wonders', Einstein published four papers of ground-breaking importance. First he published the work that introduced quanta of energy - a core idea of quantum theory. Next was a paper on Brownian motion explaining the movement of small particles suspended in a liquid. His third paper introduced the special theory of relativity linking time, distance, mass and energy while his fourth paper contains one of the most famous equations of all, E=mc².
54:56 Hamilton, Boole and their Algebras 
Professor Flood gives a fabulous overview of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and...
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
44:50 Charles Babbage and Ada Lovelace
Professor Flood explains the lives and work of the ‘Mother and Father of Computing’: http://www.gresham.ac.uk/lectures-and...
The central figure of 19th-century computing was Charles Babbage (1791-1871), who may be said to have pioneered the modern computer age with his 'difference engines' and his 'analytical engine', although his influence on subsequent generations is hard to assess. Ada, Countess of Lovelace (1815-1852), daughter of Lord Byron and a close friend of Babbage, produced a perceptive commentary on the powers and potential of the analytical engine; this was essentially an introduction to what we now call programming.
10 54:55 Gauss and Germain 
Two of the greatest mathematicians have their shared history and correspondence examined: http://www.gresham.ac.uk/lectures-and...
Carl Friedrich Gauss (1777-1855) was one of the greatest mathematicians of all time. Possibly his most famous work was his book on number theory, published in 1801. After reading this book, the French mathematicians Sophie Germain (1776-1831) began corresponding with Gauss about Fermat's last theorem, using a male pseudonym. Subsequently her interests moved to working on a general theory of vibrations of a curved surface which provided the basis for the modern theory of elasticity.
11 50:48 Hardy, Littlewood, Ramanujan and Cartwright
The story of the most productive collaborations in mathematical history: http://www.gresham.ac.uk/lectures-and...
The collaboration between G.H. Hardy (1877-1947) and J.E. Littlewood (1885-1977) was the most productive in mathematical history. Dominating the English mathematical scene for the first half of the 20th century, they obtained results of great influence, most notably in analysis and number theory. Into their world came the brilliant and intuitive mathematician, Srinivasa Ramanujan (1887-1920), who left India to work with Hardy until his untimely death at the age of 32.
12 52:11 Turing and von Neumann
An overview of the major contributions of two of the founders of computer science - John von Neumann and Alan Turing http://www.gresham.ac.uk/lectures-and...
Alan Turing (1912-1954) and John von Neumann (1903-1957) had an enormous range of interests not only in pure mathematics but also in practical applications. They made major contributions during the Second World War; Turing on cryptography and von Neumann on weapons development. The Turing machine formalised the idea of an algorithm and the Turing test is important in artificial intelligence while von Neumann founded the subject of game theory. Both are considered founders of computer science.

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